How to make a particle simulation like this?
I suggest you to try in 2D with a plane and a noise texture. You could add keyframes to change the texture over time and make it move like in your example. Then multiply it to another noise texture (with a MixRGB node) to create this noisy particle effect. Otherwise if you want to go with particles : create the scope with a simple shape and collision property, remove gravity from particles, and add a turbulence force in the scene. Then rotate both emitter and turbulence.